Improving the minimum distance bound of Trace Goppa codes

Abstract

In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form $g(x)=x+x^q+\cdots +x^{q^{m-1}}$ where $m\ge 3$ (i.e. $g(x)$ is a trace polynomial from a field extension of degree $m\ge 3$) has a better minimum distance than what the Goppa bound $d\ge 2deg(g(x))+1$ implies. Our improvement is based on finding another Goppa polynomial $h$ such that $C(L,g)=C(M,h)$ but $deg(h)>deg(g)$. This is a significant improvement over Trace Goppa codes over quadratic field extensions (i.e. the case $m=2$), as the Goppa bound for the quadratic case is sharp.